9.1: Inertial Motion

To begin our study of currents near the sea surface, let’s consider first a very simple solution to the equations of motion, the response of the ocean to an impulse that sets the water in motion. For example, the impulse can be a strong wind blowing for a few hours. The water then moves only under the influence of Coriolis force. No other force acts on the water.

Such motion is said to be inertial. The mass of water continues to move due to its inertia. If the water were in space, it would move in a straight line according to Newton’s second law. But on a rotating Earth, the motion is much different.

From Section 7.6 the equations of motion for a parcel of water moving in the ocean without friction are: \[\begin{align} \frac{du}{dt} &= -\frac{1}{\rho} \frac{\partial p}{\partial x} + 2 \Omega \ v \sin \varphi \\ \frac{dv}{dt} &= -\frac{1}{\rho} \frac{\partial p}{\partial y} - 2 \Omega \ u \sin \varphi \\ \frac{dw}{dt} &= -\frac{1}{\rho} \frac{\partial p}{\partial z} + 2 \Omega \ u \cos \varphi - g \end{align} \nonumber \]

where \(p\) is pressure, \(\Omega = 2 \pi/\text{(sidereal day)} = 7.292 \times 10^{-5} \ \text{rad/s}\) is the rotation of the Earth in fixed coordinates, and \(\varphi\) is latitude.

Let’s now look for simple solutions to these equations. To do this we must simplify the momentum equations. First, if only the Coriolis force acts on the water, there must be no horizontal pressure gradient: \[\frac{\partial p}{\partial x} = \frac{\partial p}{\partial y} = 0 \nonumber \]

Furthermore, we can assume that the flow is horizontal, and \((\PageIndex{1}) - (\PageIndex{3})\) become: \[\begin{align} \frac{du}{dt} &= 2 \Omega \ v \sin \varphi = fv \\ \frac{dv}{dt} &= -2 \Omega \ u \sin \varphi = -fu \end{align} \nonumber \]

where: \[\boxed{f = 2 \Omega \sin \varphi} \nonumber \]

\(f\) is the Coriolis Parameter and \(\Omega = 7.292 \times 10^{-5}/\text{s}\) is the rotation rate of earth. Equations \((\PageIndex{3}) - (\PageIndex{4})\) are two coupled, first-order linear differential equations which can be solved with standard techniques. If we solve the second equation for \(u\) and insert it into the first equation we obtain: \[\frac{du}{dt} = -\frac{1}{\rho} \frac{d^{2} v}{dt^{2}} = fv \nonumber \]

Rearranging the equation puts it into a standard form we should recognize, the equation for the harmonic oscillator: \[\frac{d^{2}v}{dt^{2}} + f^{2} v = 0 \nonumber \]

which has the solution \((\PageIndex{8})\). This current is called an inertial current or inertial oscillation: \[\boxed{\begin{array}{l} u &= V \sin ft \\ v &= V \cos ft \\ V^{2} &= u^{2} + v^{2} \end{array}} \nonumber \]

Note that equations \((\PageIndex{8})\) are the parametric equations for a circle with diameter \(D_{i} = 2V/f\) and period \(T_{i} = (2 \pi)/f = T_{sd}/(2 \sin \varphi)\) where \(T_{sd}\) is a sidereal day.

\(T_{i}\) is the inertial period. It is one half the time required for the rotation of a local plane on earth’s surface (Table \(\PageIndex{1}\)). The rotation is anti-cyclonic: clockwise in the northern hemisphere, counterclockwise in the southern. Inertial currents are the free motion of parcels of water on a rotating plane.

Inertial currents are the most common currents in the ocean (figure \(\PageIndex{1}\)). Webster (1968) reviewed many published reports of inertial currents and found that currents have been observed at all depths in the ocean and at all latitudes. The motions are transient and decay in a few days. Oscillations at different depths or at different nearby sites are usually incoherent.

Inertial currents are caused by rapid changes of wind at the sea surface, with rapid changes of strong winds producing the largest oscillations. Although we have derived the equations for the oscillation assuming frictionless flow, friction cannot be completely neglected. With time, the oscillations decay into other surface currents. (See, for example, Apel, 1987: §6.3 for more information.)

Figure \(\PageIndex{1}\): Inertial currents in the North Pacific in October 1987 (days 275–300) measured by holey-sock drifting buoys drogued at a depth of 15 meters. Positions were observed 10–12 times per day by the Argos system on NOAA polar-orbiting weather satellites and interpolated to positions every three hours. The largest currents were generated by a storm on day 277. Note these are not individual eddies. The entire surface is rotating. A drogue placed anywhere in the region would have the same circular motion. After van Meurs (1998).